a)
[tex](x-2)^3+7x^2+(2+x)(x^2-2x+4) > 2x^3+12x+25\\x^3-6x^2+12x-8+7x^2+2x^2-4x+8+x^3-2x^2+4x > 2x^3+12x+25\\x^3+x^3-2x^3-6x^2+7x^2+2x^2-2x^2+12x-4x+4x-12x-8+8-25 > 0\\x^2-25 > 0\\x^2 > 25\\x > 5 lubx < -5[/tex]
x∈(-∞;-5)∪(5;+∞)
b)
[tex](x-4)(x^2+4x+16)-2(x-1)^3 > =5(x-4)(x+3)-x^3\\x^3-64-2x^3+6x^2-6x+2 > =5x^2+15x-20x-60-x^3\\x^3-2x^3+x^3+6x^2-5x^2-6x-15x+20x-64+2+60 > =0\\x^2-x-2 > =0\\delta=b^2-4ac=(-1)^2-4*1*(-2)=1+8=9\\\\\sqrt{delta}=\sqrt{9}=3\\x_1=\frac{-b-\sqrt{delta}}{2a}=\frac{1-3}{2}=-1\\x_2=\frac{-b+\sqrt{delta}}{2a}=\frac{1+3}{2}=2\\\\[/tex]
Rysunek w załączniku
x∈(-∞;-1>∪<2;+∞)
c)
[tex]x^2(x-5)-(x-2)^3 < =3x-12(x-1)\\x^3-5x^2-x^3+6x^2-12x+8 < =3x-12x+12\\x^3-x^3-5x^2+6x^2-12x-3x+12x+8-12 < =0\\x^2-3x-4 < =0\\delta=b^2-4ac=(-3)^2-4*1*(-4)=9+16=25\\\sqrt{delta}=\sqrt{25}=5\\x_1=\frac{-b-\sqrt{delta}}{2a}=\frac{3-5}{2}=-1\\x_2=\frac{-b+\sqrt{delta}}{2a}=\frac{3+5}{2}=4\\\\[/tex]
Rysunek w załączniku
x∈<-1;4>
d)
[tex](3+x)^3+(2x-1)(4x^2+2x+1)-11(3x+4) < 9x^2(x+1)-(x^2+2)\\27+27x+9x^2+x^3+8x^3-1-33x-44 < 9x^3+9x^2-x^2-2\\x^3+8x^3-9x^3+9x^2-9x^2+x^2+27x-33x+27-1-44+2 < 0\\x^2-6x-16 < 0\\delta=b^2-4ac=(-6)^2-4*1*(-16)=36+64=100\\\sqrt{delta}=\sqrt{100}=10\\x_1=\frac{-b-\sqrt{delta}}{2a}=\frac{6-10}{2}=-2\\x_2=\frac{-b+\sqrt{delta}}{2a}=\frac{6+10}{2}=8[/tex]
Rysunek w załączniku
x∈(-2;8)
